<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Integrating factor</dfn> It is sometimes possible to convert a differential equation not exact into an exact equation by multiplying the equation with a suitable integrating factor. Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_30.html">
\begin{equation*}
M(x, y)+N(x, y)\frac{\textrm{d} y}{\textrm{d} x}=0.
\end{equation*}
</div>
<p class="continuation">Suppose that it is not exact. Now, we multiplying it by an integrating factor <span class="process-math">\(u(x, y)\)</span> which gives</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_30.html">
\begin{equation*}
u M+u N \frac{\textrm{d} y}{\textrm{d} x}=0.
\end{equation*}
</div>
<p class="continuation">The purpose is trying to make this new equation to be exact, i. e.,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_30.html">
\begin{equation}
\frac{\partial (u M)}{\partial y}=\frac{\partial (u N)}{\partial x}~\rightarrow~M \frac{\partial u}{\partial y}-N \frac{\partial u}{\partial x}+u \left( \frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}    \right)=0.\tag{2.6.7}
\end{equation}
</div>
<p class="continuation">But (<a href="" class="xref" data-knowl="./knowl/eq2_30.html" title="Equation 2.6.7">(2.6.7)</a>) is not so easy to be solved, except in special cases.</p>
<span class="incontext"><a href="sec2_6.html#p-51" class="internal">in-context</a></span>
</body>
</html>
